Continuous random variable

A continuous random variable is a random variable whose
cumulative distribution function is a
continuous function. The set of values it can take is not countable and these
values, when taken one by one, have zero probability of being observed.

Synonyms

Continuous random variables are sometimes also called absolutely
continuous.

Definition

The following is a formal definition.

Definition
A random variable
is said to be continuous if the probability that it assumes a value in a given
interval
can be expressed as an
integral:where
the integrand function
is called the probability density function of
.

Note that, as a consequence of this definition, the cumulative distribution
function of
isBecause
integrals are continuous with respect to their upper bound of integration,
is continuous in
,
which explains why we have stated above that a continuous random variable is a
random variable whose cumulative distribution function is continuous.

Example

Let
be a continuous random variable that can take any value in the interval
.
Let its probability density function
beThen,
for example, the probability that
takes a value between
and
can be computed as
follows:

More details

Continuous random variables are discussed in more detail in the lecture
entitled Random variables.

You can also read a brief introduction to the probability density function,
including some examples, in the glossary entry entitled
Probability density function.